Accumulation point

Accumulation point
In mathematics, a limit point, accumulation point, or cluster point of a set in a topological space is a point that can be "approximated" by points of in the sense that every neighbourhood of with respect to the topology on also contains a point of other than itself. A limit point of a set does not itself have to be an element of
There is also a closely related concept for sequences. A cluster point or accumulation point of a sequence in a topological space is a point such that, for every neighbourhood of there are infinitely many natural numbers such that This definition of a cluster or accumulation point of a sequence generalizes to nets and filters.
The similarly named notion of a (respectively, a limit point of a filter, a limit point of a net) by definition refers to a point that the sequence converges to (respectively, the filter converges to, the net converges to). Importantly, although "limit point of a set" is synonymous with "cluster/accumulation point of a set", this is not true for sequences (nor nets or filters). That is, the term "limit point of a sequence" is synonymous with "cluster/accumulation point of a sequence".
The limit points of a set should not be confused with adherent points (also called ) for which every neighbourhood of contains some point of . Unlike for limit points, an adherent point of may have a neighbourhood not containing points other than itself. A limit point can be characterized as an adherent point that is not an isolated point.
Limit points of a set should also not be confused with boundary points. For example, is a boundary point (but not a limit point) of the set in with standard topology. However, is a limit point (though not a boundary point) of interval in with standard topology (for a less trivial example of a limit point, see the first caption).
This concept profitably generalizes the notion of a limit and is the underpinning of concepts such as closed set and topological closure. Indeed, a set is closed if and only if it contains all of its limit points, and the topological closure operation can be thought of as an operation that enriches a set by uniting it with its limit points.
Definition
Accumulation points of a set
Let be a subset of a topological space
A point in is a limit point or cluster point or if every neighbourhood of contains at least one point of different from itself.
It does not make a difference if we restrict the condition to open neighbourhoods only. It is often convenient to use the "open neighbourhood" form of the definition to show that a point is a limit point and to use the "general neighbourhood" form of the definition to derive facts from a known limit point.
If is a space (such as a metric space), then is a limit point of if and only if every neighbourhood of contains infinitely many points of In fact, spaces are characterized by this property.
If is a Fréchet–Urysohn space (which all metric spaces and first-countable spaces are), then is a limit point of if and only if there is a sequence of points in whose limit is In fact, Fréchet–Urysohn spaces are characterized by this property.
The set of limit points of is called the derived set of
Special types of accumulation point of a set
If every neighbourhood of contains infinitely many points of then is a specific type of limit point called an of
If every neighbourhood of contains uncountably many points of then is a specific type of limit point called a condensation point of
If every neighbourhood of is such that the cardinality of
equals the cardinality of then is a specific type of limit point called a of
Accumulation points of sequences and nets
In a topological space a point is said to be a or if, for every neighbourhood of there are infinitely many such that
It is equivalent to say that for every neighbourhood of and every there is some such that
If is a metric space or a first-countable space (or, more generally, a Fréchet–Urysohn space), then is a cluster point of if and only if is a limit of some subsequence of
The set of all cluster points of a sequence is sometimes called the limit set.
Note that there is already the notion of limit of a sequence to mean a point to which the sequence converges (that is, every neighborhood of contains all but finitely many elements of the sequence). That is why we do not use the term of a sequence as a synonym for accumulation point of the sequence.
The concept of a net generalizes the idea of a sequence. A net is a function where is a directed set and is a topological space. A point is said to be a or if, for every neighbourhood of and every there is some such that equivalently, if has a subnet which converges to Cluster points in nets encompass the idea of both condensation points and ω-accumulation points. Clustering and limit points are also defined for filters.
Relation between accumulation point of a sequence and accumulation point of a set
Every sequence in is by definition just a map so that its image can be defined in the usual way.
If there exists an element that occurs infinitely many times in the sequence, is an accumulation point of the sequence. But need not be an accumulation point of the corresponding set For example, if the sequence is the constant sequence with value we have and is an isolated point of and not an accumulation point of
If no element occurs infinitely many times in the sequence, for example if all the elements are distinct, any accumulation point of the sequence is an -accumulation point of the associated set
Conversely, given a countable infinite set in we can enumerate all the elements of in many ways, even with repeats, and thus associate with it many sequences that will satisfy
Any -accumulation point of is an accumulation point of any of the corresponding sequences (because any neighborhood of the point will contain infinitely many elements of and hence also infinitely many terms in any associated sequence).
A point that is an -accumulation point of cannot be an accumulation point of any of the associated sequences without infinite repeats (because has a neighborhood that contains only finitely many (possibly even none) points of and that neighborhood can only contain finitely many terms of such sequences).
Properties
Every limit of a non-constant sequence is an accumulation point of the sequence.
And by definition, every limit point is an adherent point.
The closure of a set is a disjoint union of its limit points and isolated points ; that is,
A point is a limit point of if and only if it is in the closure of
If we use to denote the set of limit points of then we have the following characterization of the closure of : The closure of is equal to the union of and This fact is sometimes taken as the of closure.
A corollary of this result gives us a characterisation of closed sets: A set is closed if and only if it contains all of its limit points.
No isolated point is a limit point of any set.
A space is discrete if and only if no subset of has a limit point.
If a space has the trivial topology and is a subset of with more than one element, then all elements of are limit points of If is a singleton, then every point of is a limit point of
See also
Citations
References
Topology
General topology